Question

which points have the same distance between them as points a and b shown on the coordinate plane? select all that apply.

Explanation:

1. First, assume the coordinates of point \(A=(1,4)\) and point \(B = (- 2,-1)\) (by observing the grid - based on the standard \(x - y\) coordinate system where the horizontal axis is \(y\) and the vertical axis is \(x\)).

• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• For points \(A(1,4)\) and \(B(-2,-1)\), we have \(x_1 = 1,y_1 = 4,x_2=-2,y_2=-1\).
• Calculate \((x_2 - x_1)\) and \((y_2 - y_1)\):
• \(x_2 - x_1=-2 - 1=-3\).
• \(y_2 - y_1=-1 - 4=-5\).
• Then, \(d=\sqrt{(-3)^2+(-5)^2}=\sqrt{9 + 25}=\sqrt{34}\).

1. To find other pairs of points with the same distance, we need to check different pairs of points on the coordinate - plane using the distance formula for each pair. However, since no other points are given in the problem statement, if we were to generate points \((x,y)\) and \((m,n)\) and set \(\sqrt{(x - m)^2+(y - n)^2}=\sqrt{34}\), then \((x - m)^2+(y - n)^2 = 34\).

• For example, if we consider two points \((x_1,y_1)\) and \((x_2,y_2)\) such that \(|x_2 - x_1| = 3\) and \(|y_2 - y_1| = 5\) (or vice - versa), the distance between them will be \(\sqrt{3^2+5^2}=\sqrt{34}\).
• Let's assume a general point - finding approach. If we start from a point \((a,b)\) and move 3 units in the \(x\) - direction and 5 units in the \(y\) - direction (or vice - versa) to get another point \((a\pm3,b\pm5)\) (or \((a\pm5,b\pm3)\)), the distance between the two points will be \(\sqrt{34}\).

Since no other points are provided in the problem to select from, we can't give a specific list of points as an answer. But the general method to find points with the same distance as \(A\) and \(B\) is to use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) and find pairs of points \((x_1,y_1)\) and \((x_2,y_2)\) such that \((x_2 - x_1)^2+(y_2 - y_1)^2 = 34\).

If we assume we want to find points in a grid - based system around \(A\) and \(B\):

• Starting from point \(A(1,4)\), points like \((4,9)\) (move 3 units right and 5 units up: \(x=1 + 3=4,y = 4+5 = 9\)), \((-2,-1)\) (already \(B\)), \((-4,3)\) (move 5 units left and 1 unit up: \(x=1-5=-4,y = 4 + 1=3\)) etc. would have the same distance.

Since the problem is incomplete (no list of points to select from), we can't give a proper "Answer" section as required. But the steps to solve it are as above. If we had a set of points \((x_i,y_i)\) to check against, we would calculate the distance between each pair of points using the distance formula \(d=\sqrt{(x_j - x_i)^2+(y_j - y_i)^2}\) for all \(i
eq j\) and compare it to the distance between \(A\) and \(B\) (which is \(\sqrt{34}\)).

If we assume we are just showing the process for a general case:

Step1: Identify coordinates of \(A\) and \(B\)

Let \(A=(1,4)\) and \(B=(-2,-1)\)

Step2: Calculate distance between \(A\) and \(B\)

\(d=\sqrt{(-2 - 1)^2+(-1 - 4)^2}=\sqrt{(-3)^2+(-5)^2}=\sqrt{9 + 25}=\sqrt{34}\)

Step3: Generalize for other points

For two points \((x_1,y_1)\) and \((x_2,y_2)\), if \((x_2 - x_1)^2+(y_2 - y_1)^2 = 34\), they have the same distance as \(A\) and \(B\)

Answer:

No specific answer as no points to select from are given. General method is to use \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) and check if \(d = \sqrt{34}\) for pairs of points.